If \(\delta \)  is zero in \(U\)  then \({\tau _{\text{con}}}(p,\cdot )\)  is infinite and we are done. So we suppose that \(\delta \)  equals to \(k\in \lbrace 1,\ldots ,n-1\rbrace \)  in \(U\)  and get \({\tau _{\text{con}}}(p,v)<\infty \)  for every \(v \in U\) .
Let \(\xi _1,\ldots ,\xi _{n-1}\)  be a base of \(T_{v_0}S_pM\)  and use the formula
\(J_{v_0,\beta }(t)=\mathrm {D}\exp _{p}\bigg |_{tv_0} t\xi _\beta , \quad \text{ for } \beta \in \lbrace 1,\ldots ,n-1\rbrace ,\) 
from [1]}, to define \((n-1)\) -Jacobi fields \(J_{v_0,1}(t),\ldots , J_{v_0,n-1}(t)\)  along the geodesic \(\gamma _{p,v_0}\) . They span the vector space of all Jacobi fields along \(\gamma _{v_0}(t)\)  that vanishes at \(t=0\)  and are normal to \(\dot{\gamma }_{v_0}(t)\) .
As \(\delta (v_0)=k\)  we may assume that \(J_{v_0,\beta }({\tau _{\text{con}}}(p,v_0))=0\)  for \(\beta \in \lbrace 1,\ldots ,k\rbrace \) , implying \(D_tJ_{v_0,\beta }({\tau _{\text{con}}}(p,v_0))\ne 0\)  and \(J_{v_0,\alpha }({\tau _{\text{con}}}(p,v_0))\ne 0\)  for \(\beta \in \lbrace 1,\ldots ,k\rbrace \)  and \(\alpha \in \lbrace k+1,\ldots ,n-1\rbrace \) . Moreover, the vectors
\(D_tJ_{v_0,1}({\tau _{\text{con}}}(p,v_0)),\ldots , D_tJ_{v_0,k}({\tau _{\text{con}}}(p,v_0)), J_{v_0,k+1}({\tau _{\text{con}}}(p,v_0)), \ldots J_{v_0,n-1}({\tau _{\text{con}}}(p,v_0))\) 
are linearly independent due to [2]} and the properties of the geodesic flow on \(SN\)  presented in [3]}.
Since Jacobi fields are solutions of the second order ODE they depend smoothly on the coefficients of the respective equation. In particular after choosing smaller \(U\)  if necessary we can construct family of Jacobi fields \(J_{v,1}(t),\ldots , J_{v,n-1}(t)\)  along the geodesic \(\gamma _{p,v}(t)\)  that depend smoothly on \(v \in U\) , and span the vector space of all Jacobi fields along \(\gamma _{p,v}(t)\)  that vanishes at \(t=0\)  and are normal to \(\dot{\gamma }_{p,v}(t)\) .
Therefore the function
\(f\colon U\times [0,{\tau _{\text{con}}}(p,v_0)+1]\rightarrow {\bf R}, \quad f(v,t)=\det (J_{v,1}(t),\ldots , J_{v,n-1}(t)),\) 
is smooth and vanishes at \((v,t)\)  if and only if \(\gamma _{p,v}(t)\)  is conjugate to \(p\) .
We choose a parallel frame \(E_1(t),\ldots ,E_{n-1}(t)\)  along \(\gamma _{p,v_0}(t)\)  that is orthogonal to \(\dot{\gamma }_{p,v_0}(t)\) . With respect to this frame we write
\(J_{v_0,\beta }(t)=j_{\beta }^\alpha (t)E_{\alpha }(t), \quad \hbox{ for } \alpha , \beta \in {1,\ldots ,n-1},\) 
for some some smooth functions \(j_\beta ^\alpha (t)\) . From here we get
\(\frac{\partial ^{j}}{\partial t^{j}}f(v_0,t_0)=\frac{\partial ^{j}}{\partial t^{j}}\left(\sum _{\sigma } \text{sign}(\sigma ) j_1^{\sigma (1)}(t)j_2^{\sigma (2)}(t)\cdots j_{n-1}^{\sigma (n-1)}(t) \right)\bigg |_{t={\tau _{\text{con}}}(p,v_0)}=0,\) 
\(\hbox{for every } j\in \lbrace 0,\ldots ,k-1\rbrace \) . Above \(\sigma \)  is a permutation of the set \(\lbrace 1,\ldots ,n-1\rbrace \) . Moreover the covariant derivative of \(J_{v_0,\beta }\)  along \(\gamma _{p,v_0}\)  writes as
\(D_tJ_{v_0,\beta }(t)= \left(\frac{\mathrm {d}}{\mathrm {d}t}j_{\beta }^\alpha (t)\right)E_{\alpha }(t).\)
If \(A\)  is the square matrix whose column vectors are given in the formula (REF ) we have
\(\begin{split}&\frac{\partial ^{k}}{\partial t^{k}}f(v_0,{\tau _{\text{con}}}(p,v_0))= k!\det (A)\ne 0.\end{split}\) 
Since \(\delta (\cdot )\)  is constant \(k\)  in the set \(U\)  we have that \(\frac{\partial ^{k-1}}{\partial t^{k-1}}f(v,{\tau _{\text{con}}}(p,v))=0\)  for every \(v \in U\) . Therefore the Implicit function theorem gives that the conjugate distance function is smooth in some neighborhood \(V\subset U\)  of \(v_0\) . Since \(v_0\in U\)  was chosen arbitrarily the claim follows.
